nLab rational zero theorem

Definition

For polynomial functions

The rational zero theorem or rational root theorem states:

Given a natural number $n$ and a degree $n$ polynomial function $f \colon \mathbb{Q} \to \mathbb{Q}$ on the rational numbers with integers valued coefficients $a \colon [0,n] \to \mathbb{Z} \hookrightarrow \mathbb{Q}$ and $a_n \neq 0$, defined as

$$f(x) \coloneqq \sum_{i = 0}^{n} a_i \, x^i \,,$$

then the fiber of $f$ at $0$ is inhabited if one of the following is true:

• $a_0 = 0$,

• $a_0 \neq 0$ and there exists integers $m$ and $p$ such that $gcd(\vert m \vert, \vert p \vert) = 1$, $m \vert a_0$, and $p \vert a_n$.

For polynomials

The rational root theorem for polynomials states:

Given a natural number $n$ and a degree $n$ univariate polynomial on the rational numbers $a:\mathbb{Q}[x]$ where $a_{n} = 1$, there exists a degree $1$ univariate polynomial $b:\mathbb{Q}[x]$ where $b_{1} = 1$ such that $b | a$ if and only if there exists an integer $m$ such that $gcd(\vert m \vert, \vert b_0 \vert) = 1$ and $m \cdot a_0 = b_0$.

See also

• Gauss's lemma for polynomials?
• polynomial function
• rational numbers
• integers

References

• Wikipedia, Rational zero theorem